Solution of Inhomogeneous Helmholtz Equation

The inhomogeneous Helmholtz wave equation is conveniently solved by means of a Green's function, $G_\omega({\bf r}, {\bf r}')$ , that satisfies

$$(\nabla^{\,2} + k^{\,2})\,G_\omega ({\bf r}, {\bf r}')= -\delta({\bf r} - {\bf r}').$$ (1506)

The solution of this equation, subject to the Sommerfeld radiation condition, which ensures that sources radiate waves instead of absorbing them, is written

$$G_\omega ({\bf r}, {\bf r}') =\frac{{\rm e}^{\,{\rm i}\,k\,\vert{\bf r} -\bf {r}'\vert}} {4\pi \,\vert{\bf r}-{\bf r}'\vert}.$$ (1507)

(See Chapter 1.)

As is well known, the spherical harmonics satisfy the completeness relation

$$\sum_{l=0,\infty} \sum_{m=-l,+l}Y^{\,\ast}_{lm}(\theta',\varphi')... ...m}(\theta,\varphi) = \delta(\varphi-\varphi')\,\delta(\cos\theta -\cos\theta').$$ (1508)

Now, the three-dimensional delta function can be written

$$\delta({\bf r}-{\bf r}') = \frac{\delta(r-r')}{r^{\,2}}\, \delta(\varphi-\varphi')\, \delta(\cos\theta-\cos\theta').$$ (1509)

It follows that

$$\delta({\bf r}-{\bf r}') = \frac{\delta(r-r')}{r^{\,2}}\sum_{l=0,\infty}\sum_{m=-l,+l} Y^{\,\ast}_{lm}(\theta',\varphi')\, Y_{lm}(\theta,\varphi).$$ (1510)

Let us expand the Green's function in the form

$$G_\omega({\bf r}, {\bf r}') =\sum_{l,m} g_l(r, r') \,Y_{lm}^{\,\ast}(\theta',\varphi') \,Y_{lm}(\theta,\varphi).$$ (1511)

Substitution of this expression into Equation (1508) yields

$$\left[\frac{d^{\,2}}{dr^{\,2}} +\frac{2}{r}\frac{d}{dr}+k^{\,2} -\frac{l\,(l+1)}{r^{\,2}}\right] g_l = - \frac{\delta(r-r')}{r^{\,2}}.$$ (1512)

The appropriate boundary conditions are that $g_l(r)$ be finite at the origin, and correspond to an outgoing wave at infinity (i.e., $g\propto {\rm e}^{\,{\rm i}\,k\,r}$ in the limit $r\rightarrow \infty$ ). The solution of the above equation that satisfies these boundary conditions is

$$g_l(r, r') = A \,j_l(k\,r_<)\, h_l^{(1)}(k \,r_>),$$ (1513)

where $r_<$ and $r_>$ are the greater and the lesser of $r$ and $r'$ , respectively. The appropriate discontinuity in slope at $r=r'$ is assured if $A = {\rm i}\,k$ , because

$$\frac{d h_l^{(1)}(z)}{dz}\, j_l (z) - h_l^{(1)}(z)\, \frac{d j_l(z)}{dz} = \frac{{\rm i}} {z^{\,2}}.$$ (1514)

Thus, the expansion of the Green's function becomes

$$\frac{{\rm e}^{\,{\rm i}\,k\,\vert{\bf r} -\bf {r}'\vert}} {4\pi ... ...,r_>)\sum_{m=-l,+l} Y_{lm}^{\,\ast}(\theta',\varphi') \,Y_{lm}(\theta,\varphi).$$ (1515)

This is a particularly useful result, as we shall discover, because it easily allows us to express the general solution of the inhomogeneous wave equation as a multipole expansion.